Speaker:  Levent Alpöge (Harvard University)

Title:  A "height-free" effective isogeny estimate for abelian varieties of GL₂-type.

Abstract:  Let g ∈ ℤ^+, K a number field, S a finite set of places of K, and A, B/K g-dimensional abelian varieties with good reduction outside S which are K-isogenous and of GL₂​-type over ℚ. We show that there is a K-isogeny A → B of degree effectively bounded in terms of g, K, and S only.

We deduce among other things an effective upper bound on the number of S-integral K-points on a Hilbert modular variety.

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